Chain-center duality for locally compact groups.

AbstractThe chain group <italic>Ch</italic>(<italic>G</italic>) of a locally compact group <italic>G</italic> has one generator gρ for each irreducible unitary <italic>G</italic>-representation <italic>ρ</italic>, a relation gρ=gρ′gρ″ whenever <italic>ρ</italic> is weakly contained in ρ′⊗ρ″, and gρ*=gρ−1 for the representation ρ* contragredient to <italic>ρ. G</italic> satisfies chain-center duality if assigning to each gρ the central character of <italic>ρ</italic> is an isomorphism of <italic>Ch</italic>(<italic>G</italic>) onto the dual Z(G)̂ of the center of <italic>G</italic>. We prove that <italic>G</italic> satisfies chain-center duality if it is (a) a compact-by-abelian extension, (b) connected nilpotent, (c) countable discrete icc or (d) connected semisimple; this generalizes M. Müger’s result compact groups satisfy chain-center duality. [ABSTRACT FROM AUTHOR]